Nash implementation of matching rules

We consider the Nash implementation of Pareto optimal and individually rational solutions in the context of matching problems. We show that all such rules are supersolutions of the stable rule. Among these solutions, we show that the ''lower bound'' stable rule and the ''upper bound'' Pareto and individually rational rule are Nash implementable. The proofs of these results are by means of a recent technique developed by Danilov [2]. Two corollaries of interest are the stable rule is the minimal implementable solution that is Pareto optimal and individually rational and the stable rule is the minimal Nash implementable extension of any of its subsolutions. (C) 1996 Academic Press, Inc.